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In dual bases, the matrix of $T^{*}$ is the transpose of the matrix of $T$

Statement

Let $T : V \to W$ be linear between finite-dimensional spaces, with ordered bases $V = (v_{1}, \dots, v_{n})$ and $W = (w_{1}, \dots, w_{m})$ . In the dual bases,

$[T^{*}]_{W^{*}}^{V^{*}} = ([T]_{V}^{W})^{T} .$

Facts & Assumptions

Given: The displayed map, bases, and their dual bases.

[L1]

The transpose satisfies $T^{*} (g) = g \circ T$ (The transpose or algebraic adjoint $T^{*} : W^{*} \to V^{*}$ , $T^{*} (g) = g \circ T$ , of a linear map $T : V \to W$ ).

[L2]

The dual families of the finite bases are bases of the dual spaces (The dual family of a finite basis is a basis of the dual space, with the same dimension).

[L3]

The $j$ th column of a representing matrix is the coordinate column of the image of the $j$ th basis vector (Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases).

[L4]

The transpose of an $m \times n$ matrix has $(j, i)$ entry equal to the original $(i, j)$ entry (The transpose $A^{T}$ of a matrix).

Proof

technique · entrywise comparison

1.1

Write $[T]_{V}^{W} = (a_{i j})$ , so [L3] gives $T (v_{j}) = \sum_{i = 1}^{m} a_{i j} w_{i}$ and therefore $w_{i}^{*} (T (v_{j})) = a_{i j}$ .

L2L3algebra

2.1

The $(j, i)$ entry of $[T^{*}]_{W^{*}}^{V^{*}}$ is the coefficient of $v_{j}^{*}$ in the expansion of $T^{*} (w_{i}^{*}) \in V^{*}$ along the dual basis $V^{*}$ . Since every $ϕ \in V^{*}$ satisfies $ϕ = \sum_{j} ϕ (v_{j}) v_{j}^{*}$ , that coefficient is $T^{*} (w_{i}^{*}) (v_{j})$ , which by [L1] equals $w_{i}^{*} (T (v_{j})) = a_{i j}$ .

step 1.1L1L2L3

3.1

By [L4], step 2.1 says exactly that the $n \times m$ matrix of $T^{*}$ is the transpose of the $m \times n$ matrix of $T$ . The calculation also covers $m = 0$ or $n = 0$ , where the matrices are empty rectangles.

step 2.1L4∎

Depends on

Used by

Dependency tree · two levels

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Sources

H. Pinkham, Linear Algebra, Chapter 6 (standard reference, not scraped)

In dual bases, the matrix of T∗T∗ is the transpose of the matrix of TT